Newspace parameters
Level: | \( N \) | \(=\) | \( 207 = 3^{2} \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 207.j (of order \(22\), degree \(10\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.64034147226\) |
Analytic rank: | \(0\) |
Dimension: | \(30\) |
Relative dimension: | \(3\) over \(\Q(\zeta_{22})\) |
Twist minimal: | no (minimal twist has level 23) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{22}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
10.1 | −1.59301 | − | 1.83844i | 0 | −0.272894 | + | 1.89802i | 2.69128 | + | 1.22907i | 0 | −1.33225 | + | 4.53722i | −4.26162 | + | 2.73877i | 0 | −2.02769 | − | 6.90566i | ||||||
10.2 | 0.881085 | + | 1.01683i | 0 | 0.311634 | − | 2.16747i | 2.08252 | + | 0.951056i | 0 | −2.90589 | + | 9.89656i | 7.00598 | − | 4.50247i | 0 | 0.867820 | + | 2.95552i | ||||||
10.3 | 1.38322 | + | 1.59632i | 0 | −0.0656849 | + | 0.456848i | −6.26010 | − | 2.85889i | 0 | 2.54176 | − | 8.65643i | 6.28757 | − | 4.04078i | 0 | −4.09539 | − | 13.9476i | ||||||
19.1 | −0.952843 | − | 2.08644i | 0 | −0.825860 | + | 0.953093i | −3.01510 | + | 4.69158i | 0 | −2.67922 | + | 0.385214i | −6.02773 | − | 1.76990i | 0 | 12.6616 | + | 1.82046i | ||||||
19.2 | 0.282292 | + | 0.618133i | 0 | 2.31704 | − | 2.67401i | 3.24760 | − | 5.05336i | 0 | −3.20136 | + | 0.460286i | 4.91504 | + | 1.44319i | 0 | 4.04042 | + | 0.580925i | ||||||
19.3 | 1.55977 | + | 3.41542i | 0 | −6.61275 | + | 7.63152i | −2.90325 | + | 4.51755i | 0 | 7.13192 | − | 1.02541i | −21.9686 | − | 6.45058i | 0 | −19.9577 | − | 2.86949i | ||||||
28.1 | −1.44626 | − | 0.424661i | 0 | −1.45368 | − | 0.934223i | 1.77862 | + | 0.255727i | 0 | 2.68734 | − | 1.22727i | 5.65401 | + | 6.52507i | 0 | −2.46375 | − | 1.12516i | ||||||
28.2 | 1.80062 | + | 0.528710i | 0 | −0.402313 | − | 0.258551i | −5.05070 | − | 0.726181i | 0 | −8.85488 | + | 4.04389i | −5.50346 | − | 6.35133i | 0 | −8.71046 | − | 3.97793i | ||||||
28.3 | 3.10125 | + | 0.910608i | 0 | 5.42352 | + | 3.48548i | 5.40865 | + | 0.777647i | 0 | −0.889564 | + | 0.406250i | 5.17926 | + | 5.97719i | 0 | 16.0654 | + | 7.33684i | ||||||
37.1 | −1.44626 | + | 0.424661i | 0 | −1.45368 | + | 0.934223i | 1.77862 | − | 0.255727i | 0 | 2.68734 | + | 1.22727i | 5.65401 | − | 6.52507i | 0 | −2.46375 | + | 1.12516i | ||||||
37.2 | 1.80062 | − | 0.528710i | 0 | −0.402313 | + | 0.258551i | −5.05070 | + | 0.726181i | 0 | −8.85488 | − | 4.04389i | −5.50346 | + | 6.35133i | 0 | −8.71046 | + | 3.97793i | ||||||
37.3 | 3.10125 | − | 0.910608i | 0 | 5.42352 | − | 3.48548i | 5.40865 | − | 0.777647i | 0 | −0.889564 | − | 0.406250i | 5.17926 | − | 5.97719i | 0 | 16.0654 | − | 7.33684i | ||||||
109.1 | −0.952843 | + | 2.08644i | 0 | −0.825860 | − | 0.953093i | −3.01510 | − | 4.69158i | 0 | −2.67922 | − | 0.385214i | −6.02773 | + | 1.76990i | 0 | 12.6616 | − | 1.82046i | ||||||
109.2 | 0.282292 | − | 0.618133i | 0 | 2.31704 | + | 2.67401i | 3.24760 | + | 5.05336i | 0 | −3.20136 | − | 0.460286i | 4.91504 | − | 1.44319i | 0 | 4.04042 | − | 0.580925i | ||||||
109.3 | 1.55977 | − | 3.41542i | 0 | −6.61275 | − | 7.63152i | −2.90325 | − | 4.51755i | 0 | 7.13192 | + | 1.02541i | −21.9686 | + | 6.45058i | 0 | −19.9577 | + | 2.86949i | ||||||
136.1 | −1.20995 | − | 0.777587i | 0 | −0.802326 | − | 1.75685i | 2.65558 | − | 9.04406i | 0 | −5.31379 | − | 4.60443i | −1.21408 | + | 8.44409i | 0 | −10.2457 | + | 8.87791i | ||||||
136.2 | −0.0163142 | − | 0.0104845i | 0 | −1.66150 | − | 3.63819i | −1.45779 | + | 4.96477i | 0 | −1.27914 | − | 1.10838i | −0.0220779 | + | 0.153555i | 0 | 0.0758357 | − | 0.0657120i | ||||||
136.3 | 1.94274 | + | 1.24852i | 0 | 0.553766 | + | 1.21258i | −0.682875 | + | 2.32566i | 0 | 6.72814 | + | 5.82996i | 0.876504 | − | 6.09622i | 0 | −4.23029 | + | 3.66556i | ||||||
145.1 | −1.59301 | + | 1.83844i | 0 | −0.272894 | − | 1.89802i | 2.69128 | − | 1.22907i | 0 | −1.33225 | − | 4.53722i | −4.26162 | − | 2.73877i | 0 | −2.02769 | + | 6.90566i | ||||||
145.2 | 0.881085 | − | 1.01683i | 0 | 0.311634 | + | 2.16747i | 2.08252 | − | 0.951056i | 0 | −2.90589 | − | 9.89656i | 7.00598 | + | 4.50247i | 0 | 0.867820 | − | 2.95552i | ||||||
See all 30 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.d | odd | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 207.3.j.a | 30 | |
3.b | odd | 2 | 1 | 23.3.d.a | ✓ | 30 | |
12.b | even | 2 | 1 | 368.3.p.a | 30 | ||
23.d | odd | 22 | 1 | inner | 207.3.j.a | 30 | |
69.g | even | 22 | 1 | 23.3.d.a | ✓ | 30 | |
69.g | even | 22 | 1 | 529.3.b.b | 30 | ||
69.h | odd | 22 | 1 | 529.3.b.b | 30 | ||
276.j | odd | 22 | 1 | 368.3.p.a | 30 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
23.3.d.a | ✓ | 30 | 3.b | odd | 2 | 1 | |
23.3.d.a | ✓ | 30 | 69.g | even | 22 | 1 | |
207.3.j.a | 30 | 1.a | even | 1 | 1 | trivial | |
207.3.j.a | 30 | 23.d | odd | 22 | 1 | inner | |
368.3.p.a | 30 | 12.b | even | 2 | 1 | ||
368.3.p.a | 30 | 276.j | odd | 22 | 1 | ||
529.3.b.b | 30 | 69.g | even | 22 | 1 | ||
529.3.b.b | 30 | 69.h | odd | 22 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{30} - 11 T_{2}^{29} + 78 T_{2}^{28} - 411 T_{2}^{27} + 1728 T_{2}^{26} - 6157 T_{2}^{25} + 18695 T_{2}^{24} - 49567 T_{2}^{23} + 123051 T_{2}^{22} - 279029 T_{2}^{21} + 570634 T_{2}^{20} - 1009889 T_{2}^{19} + \cdots + 38809 \)
acting on \(S_{3}^{\mathrm{new}}(207, [\chi])\).